IMO Shortlist 2006 problem A3

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Dodao/la: arhiva
2. travnja 2012.
The sequence c_{0}, c_{1}, . . . , c_{n}, . . . is defined by c_{0}= 1, c_{1}= 0, and c_{n+2}= c_{n+1}+c_{n} for n \geq 0. Consider the set S of ordered pairs (x, y) for which there is a finite set J of positive integers such that x=\sum_{j \in J}{c_{j}}, y=\sum_{j \in J}{c_{j-1}}. Prove that there exist real numbers \alpha, \beta, and M with the following property: An ordered pair of nonnegative integers (x, y) satisfies the inequality m < \alpha x+\beta y < M if and only if (x, y) \in S.

Remark: A sum over the elements of the empty set is assumed to be 0.
Izvor: Međunarodna matematička olimpijada, shortlist 2006